Hessian Matrix convex, concave, or neither?

Question

I want to find out in what range a certain function is convex. In order to find out, I calculated partial derivatives and set up a Hessian matrix. As the Hessian still has variables in it, I am not sure about the interpretation. I would like to work with the method of principal minors.

I want to make use of this theorem:

Theorem: $f(x,y)$ is convex if and only if its $n \times n$ Hessian matrix is positive semidefinite for all possible values of $(x,y)$. The Hessian is positive definite if and only if its $n=2$ leading principal minors are positive.

My $2\times 2$ matrix: $$\begin{bmatrix} 6x+4 & 7855\\7855 & 2\end{bmatrix}$$

Answer 1

The leading principal minors are

1. $6x+4$
2. The determinant of the full matrix, $(6x+4)2 - 7855^2$.

For what values of $x$ are both of these positive? Those are the values of $x$ for which your function is (strictly) convex. (Note that in principle you only need to check the determinant, because the bottom-right entry is already positive ($2$) and you can permute the rows and columns so that it becomes the top-right leading principal minor instead of $6x+4$).

In this case there is only a single point where the Hessian is semi-definite, so you don't need to deal with the semi-definite case (you can include the point in the convex set).